Issue with graphical interpretation of the universal approximation theorem
The classical version of the universal approximation theorem states that, roughly, given a continuous function $f \colon [0, 1]^n \to [0, 1]^n$, there exists a single layer neural network and a set of weights and biases such that this network approximates the given function $f$ arbitrarily well.
It doesn't say anything about how you obtain such weights: the result is entirely independent of the way you train your network, and what it says is that the set of single-layer neural networks has enough capacity, in principle, to approximate any continuous function arbitrarily well.
This kind of result is fairly common in mathematics: it is shown that a suitable object must exist, but the proof is non-constructive, i.e. it doesn't actually show you how to get that object.
You're indeed correct that the backpropagation algorithm might not be able to find suitable weights. In fact, if you fix an architecture in advance, there might not even be any weights that would lead to a neural network that is a good approximation to $f$.
Why do we care?
Universality tells us, at least in principle, that we should be able to approximate the function we want, if we pick an appropriate architecture. The XOR Problem, for example, affects a type of model called a perceptron (essentially a single neuron neural network), and says that the XOR function cannot be approximated by perceptrons. It is always useful to know the kinds of functions that can and cannot be expressed by a certain type of model.